Bilinear decompositions for the product space

Ky, Luong Dang

doi:10.1002/mana.201200101

Cited by 27 publications

(17 citation statements)

References 8 publications

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“…It is easy to show that ∈ A 1 ( ), is of uniformly upper type α, I( ) = ( ) = α, ( ) is not attainable, but I( ) is attainable. Moreover, it worths to point out that such function naturally appears in the study of the pointwise multiplier characterization for the BMO-type space on the metric space with doubling measure (see [60,61]); see also [50][51][52][53] for some other applications of such functions. Throughout the whole paper, we always assume that is a growth function as in Definition 2.2.…”

Section: Definition 22mentioning

confidence: 99%

“…Recently, in [74], the last two authors of this paper studied the Musielak-Orlicz-Hardy spaces associated with nonnegative self-adjoint operators satisfying Davies-Gaffney estimates. Furthermore, some special Musielak-OrliczHardy spaces associated with the Schrödinger operator L := −∆ + V on R , where the nonnegative potential V satisfies the reverse Hölder inequality of order /2, were studied by the third author of this paper [51][52][53] and further applied to the study of commutators of singular integral operators associated with the operator L. Very recently, the authors of this paper [12] studied the weighted Hardy space associated with nonnegative self-adjoint operators satisfying the reinforced off-diagonal estimates on R (see Assumption (B) for their definitions in the present setting), which improves those results in [17,67,74] in some sense by essentially extending the range of the considered weights. We would like to describe partly the results in [74] which may be closely related to this paper.…”

Section: Introductionmentioning

confidence: 99%

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Musielak-Orlicz-Hardy Spaces Associated with Operators Satisfying Reinforced Off-Diagonal Estimates

Bui¹,

Cao²,

Ky³

et al. 2013

Analysis and Geometry in Metric Spaces

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128

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Let be a metric space with doubling measure and L a oneto-one operator of type ω having a bounded H ∞ -functional calculus in L 2 ( ) satisfying the reinforced ( L L ) off-diagonal estimates on balls, where L ∈ [1 2) and L ∈ (2 ∞]. Let:is an Orlicz function, (· ) ∈ A ∞ ( ) (the class of uniformly Muckenhoupt weights), its uniformly critical upper type index I( ) ∈ (0 1] and (· ) satisfies the uniformly reverse Hölder inequality of order ( L /I( )) , where ( L /I( )) denotes the conjugate exponent of L /I( ). In this paper, the authors introduce a Musielak-Orlicz-Hardy space H L ( ), via the Lusin-area function associated with L, and establish its molecular characterization. In particular, when L is nonnegative self-adjoint and satisfies the Davies-Gaffney estimates, the atomic characterization of H L ( ) is also obtained. Furthermore, a sufficient condition for the equivalence between H L (R ) and the classical Musielak-Orlicz-Hardy space H (R ) is given. Moreover, for the Musielak-Orlicz-Hardy space H L (R ) associated with the second order elliptic operator in divergence form on R or the Schrödinger operator L := −∆ + V with 0 ≤ V ∈ L 1 loc (R ), the authors further obtain its several equivalent characterizations in terms of various non-tangential and radial maximal functions; finally, the authors show that the Riesz transform ∇L −1/2 is bounded from H L (R ) to the Musielak-Orlicz space L (R ) when ( ) ∈ (0 1], from H L (R ) to H (R ) when ( ) ∈ ( +1 1], and from H L (R ) to the weak MusielakOrlicz-Hardy space W H (R ) when ( ) = +1 is attainable and (· ) ∈ A 1 ( ), where ( ) denotes the uniformly critical lower type index of . KeywordsMusielak-Orlicz-Hardy space • molecule • atom • maximal function • Lusin area function • Schrödinger operator • elliptic operator • Riesz transform MSC: 42B35,

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Section: Definition 22mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Musielak-Orlicz-Hardy Spaces Associated with Operators Satisfying Reinforced Off-Diagonal Estimates

Bui¹,

Cao²,

Ky³

et al. 2013

Analysis and Geometry in Metric Spaces

Self Cite

128

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“…Moreover, the Musielak-Orlicz BMOtype space BMO ϕ (R n ) was also introduced and further proved to be the dual space of H ϕ (R n ) in [63] by using the atomic characterization of H ϕ (R n ) established in [63]. Furthermore, some interesting applications of the spaces H ϕ (R n ) and BMO ϕ (R n ) were given in [11,13,14,63,64,65,66]. Moreover, the radial and the non-tangential maximal functions characterizations, the Littlewood-Paley function characterization and the molecular characterization of H ϕ (R n ) were obtained in [69,54].…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, Song and Yan [86] studied the weighted Hardy space H 1 ω, L (R n ) associated with the Schrödinger operator L, where ω ∈ A 1 (R n ). Very recently, some special Musielak-Orlicz-Hardy spaces associated with the Schrödinger operator L := −∆ + V on R n , where the nonnegative potential V satisfies the reverse Hölder inequality of order n/2, were studied by Ky [65,66] and further applied to the study of commutators of singular integral operators associated with the operator L.…”

Section: Introductionmentioning

confidence: 99%

Musielak–Orlicz–Hardy Spaces Associated with Operators and Their Applications

Yang

2012

J Geom Anal

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Let X be a metric space with doubling measure and L a nonnegative selfadjoint operator in L 2 (X ) satisfying the Davies-Gaffney estimates. Let ϕ : X × [0, ∞) → [0, ∞) be a function such that ϕ(x, ·) is an Orlicz function, ϕ(·, t) ∈ A ∞ (X ) (the class of uniformly Muckenhoupt weights), its uniformly critical upper type index I(ϕ) ∈ (0, 1] and it satisfies the uniformly reverse Hölder inequality of order 2/[2 − I(ϕ)]. In this paper, the authors introduce a Musielak-Orlicz-Hardy space H ϕ, L (X ), by the Lusin area function associated with the heat semigroup generated by L, and a Musielak-Orlicz BMO-type space BMO ϕ, L (X ), which is further proved to be the dual space of H ϕ, L (X ) and hence whose ϕ-Carleson measure characterization is deduced. Characterizations of H ϕ, L (X ), including the atom, the molecule and the Lusin area function associated with the Poisson semigroup of L, are presented. Using the atomic characterization, the authors characterize H ϕ, L (X ) in terms of the Littlewood-Paley g * λ -function g * λ, L and establish a Hörmandertype spectral multiplier theorem for L on H ϕ, L (X ). Moreover, for the Musielak-Orlicz-Hardy space H ϕ, L (R n ) associated with the Schrödinger operator L := −∆ + V , where 0 ≤ V ∈ L 1 loc (R n ), the authors obtain its several equivalent characterizations in terms of the non-tangential maximal function, the radial maximal function, the atom and the molecule; finally, the authors show that the Riesz transform ∇L −1/2 is bounded from H ϕ, L (R n ) to the Musielak-Orlicz space L ϕ (R n ) when i(ϕ) ∈ (0, 1], and from H ϕ, L (R n ) to the Musielak-Orlicz-Hardy space H ϕ (R n ) when i(ϕ) ∈ ( n n+1 , 1], where i(ϕ) denotes the uniformly critical lower type index of ϕ.

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“…Here, A q (R n ), q ∈ [1, ∞], denotes the class of Muckenhoupt weights. Moreover, more interesting applications of these spaces were also presented in [1,3,34,4,28,27,29,30,31]. Notice that Musielak-Orlicz functions are the natural generalization of Orlicz functions which may vary in the spatial variable (see, for example, [13,14,28,39]).…”

Section: Introductionmentioning

confidence: 99%

Musielak-Orlicz BMO-type spaces associated with generalized approximations to the identity

Hou

Yang

2014

Acta. Math. Sin.-English Ser.

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Let X be a space of homogenous type and ϕ : X × [0, ∞) → [0, ∞) a growth function such that ϕ(·, t) is a Muckenhoupt weight uniformly in t and ϕ(x, ·) an Orlicz function of uniformly upper type 1 and lower type p ∈ (0, 1]. In this article, the authors introduce a new Musielak-Orlicz BMO-type space BMO ϕ A (X ) associated with the generalized approximation to the identity, give out its basic properties and establish its two equivalent characterizations, respectively, in terms of the spaces BMO ϕ A, max (X ) and BMO ϕ A (X ). Moreover, two variants of the John-Nirenberg inequality on BMO ϕ A (X ) are obtained. As an application, the authors further prove that the space BMO ϕ √ ∆ (R n ), associated with the Poisson semigroup of the Laplace operator ∆ on R n , coincides with the space BMO ϕ (R n ) introduced by L. D. Ky. A t f (x) := X a t (x, y)f (y) dµ(y). Duong and Yan [17] first introduced the suitable function set M(X ) such that, for all f ∈ M(X ) and all t, s ∈where the supremum is taken over all balls B in X and t B := r m B with r B being the radius of the ball B and m a positive constant. Duong and Yan [17] gave out some basic properties of BMO A (X ) including a variant of the John-Nirenberg inequality and further proved that the space BMO √ ∆ (R n ), associated with the Poisson semigroup of the Laplace operator ∆ on R n , and BMO(R n ) coincide with equivalent norms. Tang [45] introduced the Morrey-Campanato type spaces Lip A (α, X ) associated with the generalized approximation to the identity {A t } t>0 and established the John-Nirenberg inequality on these spaces. Furthermore, Deng, Duong and Yan [12] established a new characterization of the classical Morrey-Campanato space on R n by using an appropriate convolution to replace the minimizing polynomial of a function f in the Morrey-Campanato norm. Moreover, a similar characterization for the Morrey space on R n was also obtained by Duong, Xiao and Yan in [16]. Yang and Zhou [49] introduced some generalized approximations to the identity with optimal decay conditions in the sense that these conditions are sufficient and necessary for these generalized approximations to the identity to characterize BMO(X ), which was introduced by Long and Yang [36]. Furthermore, a new John-Nirenberg-type inequality associated with the generalized approximations to the identity on BMO(X ) was also established in [49]. Recently, Bui and Duong [6] introduced the weighted BMO space BMO A (X , ω) associated to the generalized approximations to the identity, {A t } t>0 , and also obtained the John-Nirenberg inequality on these spaces.Let X be a space of homogeneous type with degree (α 0 , n 0 , N 0 ) (see Remark 2.5 below for its definition), where α 0 , n 0 and N 0 are as in (2.7), (2.3) and (2.5) below, respectively. Let ϕ : X × [0, ∞) → [0, ∞) be a growth function such that ϕ(·, t) is a Muckenhoupt weight uniformly in t, and ϕ(x, ·) is an Orlicz function of uniformly upper type 1 and lower type p ∈ (0, 1]. Motivated by [28,17,45], in this article, we first introduce the...

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Bilinear decompositions for the product space

Cited by 27 publications

References 8 publications

Musielak-Orlicz-Hardy Spaces Associated with Operators Satisfying Reinforced Off-Diagonal Estimates

Musielak-Orlicz-Hardy Spaces Associated with Operators Satisfying Reinforced Off-Diagonal Estimates

Musielak–Orlicz–Hardy Spaces Associated with Operators and Their Applications

Musielak-Orlicz BMO-type spaces associated with generalized approximations to the identity

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